HW 3: A consistent equation Ax = b where A has more columns than rows can have a unique solution.
FALSE. If Ax = b is consistent, then it either has a unique solution or it has infinitely many solutions. Since A has more columns than rows, not every column can have a pivot. Therefore, there must be at least one free variable, so there are infinitely many solutions.
This statement is false.
A consistent equation made up of a matrix A with more columns than rows, will result in a solution with infinitely many solutions or no solution at all.
Consider the system of equations Ax = b, where A is an m x n matrix with more columns than rows, x is an n x 1 column vector, and b is an m x 1 column vector.
If det(A) is equal to zero, then the rows of A are linearly dependent, which means the system of linear equations has either no solutions or an infinite number of solutions.
If det(A) is not equal to zero, then the system has a unique solution. However, this only occurs when A is a square matrix, i.e., when it has the same number of rows and columns.
Therefore, for an equation where A has more columns than rows, a unique solution cannot exist for a consistent equation.
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